As far as I can recall the difference between a Moore and a Mealy machine is that the output of a Moore machine is solely dependent on the current state, whereas the output of a Mealy machine is also dependent on the current input. This typically means that a Mealy machine has some combinatorial logic between the sequential part and the output involving signals from the input side.
The two types of machines can be shown to be equivalent, and any machine of one type can be transformed into the other. Since you were interested in the mathematical side of things, I'm guessing this property is of interest to you.

Wikipedia has some info on the subject, and I'm sure some Google-digging will turn up a formal proof, if that's your fancy.

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Also, I don't know of any particular book to recommend you (As I'm not sure excactly what you're after, and my total knowledge on the subject is summed up above), but I'd definitively try to find out whether Springer Books has anything on the subject. When it comes to dry condensed technical learning I find them hard to beat. And of course try and look up some academic articles if you have access to such a database.

As far as I can recall the difference between a Moore and a Mealy machine is that the output of a Moore machine is solely dependent on the current state, whereas the output of a Mealy machine is also dependent on the current input.

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That would be correct

Further Moore, Mealy and Mixed type machines are Synchronous State Machines.

There's nearly a hundred pages about them in my first reference.

Mathematically the subject goes back to Turing and the Turing Machine, which, as the universal machine, is top of the heirarchy of types and from which the subsequent maths has flown.

Mathematically the subject goes back to Turing and the Turing Machine, which, as the universal machine, is top of the heirarchy of types and from which the subsequent maths has flown.

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Yes.I think mathematics on this subject should go back to Turing Machine,since both Mealy and Moore is transducers,transducers is finite state machine which is another name for finite state automata.Automata theory goes back to Turing Machine.

The book I referenced above is a classic in the automata theory,and some discrete course should teach something related to automata,lingustic,etc...